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Yang W, Ren XM. Detecting Impulses in Mechanical Signals by Wavelets.
EURASIP Journal on Advances in Signal Processing 2004, 8: 946162.
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EURASIP Journal on Applied Signal Processing 2004:8, 1156–1162c© 2004 Hindawi Publishing Corporation
Detecting Impulses in Mechanical Signals by Wavelets
W.-X. YangInstitute of Vibration Engineering, Northwestern Polytechnical University, Xi’an 710072, ChinaEmail: [email protected]
X.-M. RenInstitute of Vibration Engineering, Northwestern Polytechnical University, Xi’an 710072, ChinaEmail: [email protected]
Received 21 February 2003; Revised 17 October 2003; Recommended for Publication by Marc Moonen
The presence of periodical or nonperiodical impulses in vibration signals often indicates the occurrence of machine faults. Thisknowledge is applied to the fault diagnosis of such machines as engines, gearboxes, rolling element bearings, and so on. Thedevelopment of an effective impulse detection technique is necessary and significant for evaluating the working condition of thesemachines, diagnosing their malfunctions, and keeping them running normally over prolong periods. With the aid of wavelettransforms, a wavelet-based envelope analysis method is proposed. In order to suppress any undesired information and highlightthe features of interest, an improved soft threshold method has been designed so that the inspected signal is analyzed in a moreexact way. Furthermore, an impulse detection technique is developed based on the aforementioned methods. The effectivenessof the proposed technique on the extraction of impulsive features of mechanical signals has been proved by both simulated andpractical experiments.
Keywords and phrases: wavelet transform, envelope analysis, fault diagnosis, rolling element bearing, soft threshold.
1. INTRODUCTION
The extraction of impulsive features in vibration signals isvital for diagnosing such machines as engines, rolling ele-ment bearings, gearboxes, and so on. Researchers have de-veloped many methods for fulfilling this purpose, for ex-ample, cepstrum analysis [1], signal demodulation proce-dure [2], transmission error measurement [3], higher-ordertime-frequency analysis [4], moving window procedure [5],and envelope analysis [6]. These techniques either use a timedomain averaging procedure or adopt the classical time-frequency analyzing method that only provides constanttime/frequency resolution analysis, so they are not powerfulenough to deal with nonstationary signals. Recently, interestin the use of wavelet transforms (WTs) for processing non-stationary signals has grown [7]. Different from these con-venient methods, the WTs provide a constant frequency-to-bandwidth ratio analysis. In consequence, WTs possess finetime resolution in the high frequency ranges and excellentfrequency resolution in low frequency region. This feature ofWTs uniquely fits the requirement in failure diagnosis [8].However, the impulse detection results generated by WTs arestill not easy to be identified especially when the signal-to-noise ratio (SNR) of the detected signal is low. In view of this,a new wavelet-based impulse detection technique is studiedin this paper.
2. SUPERIORITY OF MORLET WAVELETON IMPULSE DETECTION
The wavelet transform of a signal x(t) is defined as
WTx(a, τ) = ⟨ψa,τ(t), x(t)⟩
= 1√a
∫x(t)ψ∗
(t − τa
)dt,
(1)
where WTx(a, τ) represents the wavelet transforming coeffi-cient derived from the signal x(t) when setting the scale to bea and the time shifting parameter to be τ; the asterisk standsfor complex conjugate; ψa,τ(t) the daughter wavelets of themother wavelet ψ(t), which is derived by varying both thescale factor a and the shifting parameter τ continuously. Thefactor 1/
√a is used to ensure energy preservation.
From (1), it is found that the wavelet transformWTx(a, τ) is a function of the shifting parameter τ for eachscale a. It manifests the information of x(t) at different lev-els of resolution by measuring the similarity between the sig-nal x(t) and the daughter wavelet function ψa,τ(t) at differ-ent scales. This implies that the components of the signalmay be extracted out perfectly when a wavelet function withsimilar shape as the component is employed. This is calledthe maximum matching mechanism adapted for WTs. In or-der to demonstrate this matching mechanism graphically, an
Detecting Impulses in Mechanical Signals by Wavelets 1157
Impulsive feature contained in the signal
Morlet wavelet
Daubechies wavelet
Mexican hat wavelet
(a)
An arbitrary signal with periodic impulsive features
Coefficients generated by Morlet wavelet at scale 20
Coefficients generated by Daubechies wavelet at scale 20
Coefficients generated by mexican hat wavelet at scale 20
(b)
Figure 1: Wavelet transforms of a simulated signal. (a) Impulsive feature and wavelets. (b) Signal and its wavelet transforming coefficients.
example is given in the following. Figure 1a shows a simu-lated impulsive feature usually contained in the signal andthree kinds of wavelet functions (Morlet wavelet, Daubechieswavelet, and mexican hat wavelet) which are often used inpractice. Figure 1b shows a simulated signal with periodicimpulsive features and its corresponding wavelet transform-ing coefficients derived at a scale of 20 by using Daubechieswavelet, mexican hat wavelet, and Morlet wavelet, respec-tively.
From Figure 1a, it was found that when compared toother two kinds of wavelet functions, the geometric shapeof Morlet wavelet looks more like the impulsive feature con-tained in the signal. The results shown in Figure 1b furtherdemonstrate that, using Morlet wavelet, the impulsive fea-tures of the signal can be perfectly extracted. From this exam-ple, it is known that the selection of an appropriate waveletfunction is actually a crucial work for guaranteeing the suc-cessful extraction of signal features.
The single freedom degree system subjected to an impactload may be formulated as
Md2x
dt2+ C
dx
dt+ Kx = Fδ(t), (2)
where x represents the displacement, M the concentratedmass, C the damping coefficient, and K the stiffness of thesystem, F is a constant, and
δ(t) ={
1, t = τ,0, otherwise.
(3)
The solution for (2) is
x(t) = F +Mv0
Mωde−ζωnt sin
(ωdt)
+x0(
1− ζ2)1/2 e
−ζωnt cos(ωdt − ψ
),
(4)
whereωn =√K/M, ζ = C/2Mωn,ωd = ωn
√1− ζ2, the phase
angle ψ = tan−1(ζ/√
1− ζ2), and x0 and v0 indicate the initialdisplacement and velocity of the system, respectively.
When the initial displacement and velocity of the systemare zero, that is, x0 = 0, v0 = 0, (4) can be rewritten as
x(t) = Ae−ζωnt sin(ωdt), (5)
where A = F/Mωd.Equation (5) indicates that the impulsive feature, which
is caused by external impact load, is characterized by an oscil-lation with decaying amplitude. So according to the match-ing mechanism of wavelet transform that has been provedin Figure 1, Morlet wavelet could be a more suitable waveletfunction for extracting such types of features, because Mor-let wavelet has a more similar shape to the impulsive feature.The complex Morlet wavelet function can be expressed as
ψ(t) = 1√2π
e−(t2/2)β2[cos(ωt) + j sin(ωt)
]
= 1√2π
e−(t2/2)β2e jωt.
(6)
Actually, (6) is similar to (5) in both structure and composi-tions. In addition, it is noticed from (6) that the parameterβ determines the geometric shape of Morlet wavelet. When βtends to zero, the function tends to a cosine function whichhas fine frequency resolution, and when β tends to +∞, thefunction inclines to be an impulse function and its time res-olution will be increased notably. So it is natural to expectthat the Morlet wavelet with larger β is suitable to extract im-pulses in mechanical signals. It is necessary to note that, inorder to satisfy the admissibility condition of the wavelet [9]and guarantee that the modified Morlet wavelets are always“band-pass” filters, the parameter β should not be adjusted
1158 EURASIP Journal on Applied Signal Processing
arbitrarily. This limitation seems to have been omitted by[10].
It is well known that, in the frequency domain, the im-pulse has response in the whole frequency region, while theharmonic signals have response only in a narrow band of fre-quency. This suggests that we may detect impulses by per-forming WTs in one special frequency region, where the har-monic signals have little or no response, but the impulse re-sponse is still significant. Since the time resolution of wavelettransform notably increases depending on the duration ofthe mother wavelet, the nonstationary feature (including im-pulse), in most cases, can be better revealed if the wavelettransform is carried out in the high frequency region.
Therefore, two measures may be taken for impulse detec-tion. The first is to properly adjust the shape control parame-ter β of the Morlet wavelet function, the second is to performthe WTs at a special frequency region in which the harmonicsignals have little or no response, but the inspected impulsestill has strong response.
3. ENVELOPE ANALYSIS BASED ON COMPLEXMORLET WAVELET
In the past, the envelope analysis of the signal was carried outwith the aid of the Hilbert transform. In essence, the Hilberttransform can be considered to be a filter that simply shiftsphases of all frequency components of its input by −π/2 ra-dians. x(t) and y(t) form the complex conjugate pair of ananalytic signal z(t) as
z(t) = x(t) + iy(t) = A(t)eiθ(t) (7)
withA(t) = [x(t)2+y(t)2]1/2, θ(t) = tan−1[y(t)/x(t)]. Wherei = √−1, the time-varying function A(t) is the so-calledinstantaneous envelop of the signal x(t), which extracts theslow time variation of the signal.
Because the Hilbert spectrum uses a transform ratherthan convolution as in the Fourier analysis, the practicedemonstrates that for a transient signal, the Hilbert spectrumdoes offer clearer frequency-energy decomposition than thetraditional Fourier spectrum. However, during the imple-mentation of the Hilbert transform, it deals with differentfrequency components without any distinguishing. More-over, from (7), it is found that the computation of y(t) stillrequires the knowledge of x(t) for all values of t. Thus, the“local” property of the Hilbert transform is in fact a “global”property of the signal. In view of these reasons, the complexwavelet-based envelope detection method is designed for ex-tracting the impulsive features contained in the signals.
The complex Morlet wavelet transform of a modulatedsignal x(t) can be written as
WTx(a, τ)= 1√2πa
∫x(t)e−([(t−τ)/a]2/2)β2
e[ jω(t−τ)/a]dt. (8)
Since the complex Morlet wavelet is adopted in (8), allwavelet transforming coefficients WTx derived by (8) arecomplex numbers. Similar to the Hilbert-transform-based
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (s)
−0.5
0
0.5
1
1.5
Am
plit
ude
(a)
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time (s)
5
10
15
20
u(t
)
(b)
Figure 2: Distinguishing impulses from simulated noisy signal. (a)Noisy impulse signal. (b) Envelope analysis of the noisy impulse sig-nal.
envelope analysis, the wavelet-based envelope analysis is per-formed by
u(t) =√{
Re[
WTx(a, t)]}2
+{
Im[
WTx(a, t)]}2
, (9)
where u(t) indicates the envelope analysis results.In the following, a noisy impulse-contained signal was
employed for verifying the effectiveness of this new envelopeanalysis method. Figure 2a shows the original noisy signalwith impulsive features, and Figure 2b shows its correspond-ing envelope analysis result u(t). Where SNR=1.0, β = 1, thewavelet transform is performed at frequency 400 Hz.
It can be clearly seen from Figure 2 that the envelopes ofthe signal are extracted out perfectly. Additionally, as Mor-let wavelet itself is one kind of bandpass filter, the proposedmethod has a good capacity for noise reduction. The noise inthe analyzed results is suppressed to a certain extent, so thatthe impulses in the derived results become more explicit andmore easily identified. This method will undoubtedly facili-tate the machine fault diagnosis.
However, the wavelet-based envelope analysis on its ownis not sufficient to reduce noise and highlight the interestingfeatures contained in the signals. It has been reported that us-ing the “soft-thresholding” method [10] can further enhancethis function. Hence, a new flexible soft-threshold-based de-noising method is further studied in the following section.
4. SOFT THRESHOLD
The application of threshold criterion is effective in reducingnoise and highlighting the interesting features in mechanical
Detecting Impulses in Mechanical Signals by Wavelets 1159
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Number of data
−6−4−2
0246
Sa
(a)
max[|Sa(i)|]
0
−max[|Sa(i)|]
S a×
max
(|Sa|)/
max
(Sa)
1 0.5 0
S′a
(b)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
Number of data
0.2
0.4
0.6
0.8
1
∣ ∣ S a×S′ a∣ ∣
(c)
Figure 3: Working mechanism of the new soft-threshold function. (a) Sa; (b) Soft-threshold function; (c) |Sa × S′a|.
signals [10], but how to choose an ideal threshold still re-mains an unanswered question. Here, a new soft thresholdfunction is designed for making the denoising process moreadaptive and smoother. It is
S′a(i) = e−ξ×{max[|Sa(i)|]−|Sa(i)|}2, (10)
where S′a(i) represents the relative value of the ith coefficientat scale a, Sa(i)|i=1,...,N the ith coefficient, and ξ > 0 the decayparameter. It is easy to know that the larger the value of ξ, thefaster the function decays. The working mechanism of thissoft-threshold function is as illustrated in Figure 3, where thevalue of ξ is taken to be 3. By multiplying the wavelet co-efficients Sa with the soft-threshold function S′a, the resultsderived by wavelet transform may be further purified. Thepurified results are shown in Figure 3c.
From (10), it was found that the larger the value of ξ,the faster the function decays. In other words, when ξ ap-proaches to a large value, only a few number of data that arevery close to the max[|Sa|] are retained, while most data aresuppressed. Consequently, the impulsive feature in mechan-ical signals is highlighted in this case. On the contrary, whenξ approaches a small value, more data are preserved and onlya few number of data with small values are suppressed. Thiscase is more suitable for processing harmonic signals. Obvi-ously, with the aid of adjustable ξ, the proposed soft thresh-old is more adaptive for feature extraction. Besides, this softthreshold has another merit, that is, however much the dataSa is suppressed, its relative value S′a will not be zero. This isdistinctly different from many other available threshold cri-teria [11, 12]. So, in comparison, the new soft threshold can
lead to a much smoother result. The value of S′a generatedby the new soft threshold is limited to the half-closed region(0, 1].
5. DEVELOPMENT OF THE IMPULSEDETECTION METHOD
looseness=1Based on the techniques proposed above, an ad-vanced impulse detection strategy is developed, as depictedin Figure 4.
It is necessary to note that, during the implementationof this strategy, the parameters β and ξ as well as the scalea should be selected appropriately according to the prac-tical situation of the inspected signals. Often a satisfactoryimpulse detection result can be obtained when the param-eter β is taken to be a larger one. But it should be kept inmind that it should not be adjusted arbitrarily so that the ad-missibility condition of the wavelet [9] is satisfied. WTs areused at high frequencies to detect shock impulses in signalsmeasured from rolling element bearings. However, when di-agnosing gearbox vibration, the impulses generated due totooth breakage may be identified more satisfactorily if theWTs are performed at frequencies lower than the meshingfrequency of the gear couple. In practical applications, ap-propriate scale region of WTs can be selected by using themethod given in [13].
To verify the effectiveness of the strategy, a simulated ex-periment was performed first. The raw signal and the im-pulse detection results obtained in the high frequency re-gion [0.4 kHz, 2.0 kHz] are shown in Figure 5, where β = 5.6,ξ = 6.
1160 EURASIP Journal on Applied Signal Processing
Start
Pre-denoising the raw signal
Selection of β, a, and ξ
Wavelet transform of the denoised signal by usingcomplex Morlet wavelet function
Envelope calculation
u(t) =√{
Re[
WTx(a, t)]}2
+{
Im[
WTx(a, t)]}2
Envelope analysis of the denoised signal
Purifying the envelope analysis resultby using the soft threshold
Displaying the impulse detection results
End
Figure 4: Flow chart of the impulse detection method.
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.4
0.8
1.2
1.6
2.0
Freq
uen
cy(K
Hz)
0 0.1 0.2 0.3 0.4 0.5
Time (s)
−101
Am
plit
ude
Figure 5: Detecting impulses from the noisy signal using the im-pulse detection method depicted in Figure 4.
Figure 5 suggests that the proposed strategy is actuallyeffective at impulse detection even if the inspected signal isheavily polluted by background noise. In the following, thestrategy to deal with the vibratory signals collected from aball bearing with ball flaw fault was applied. The geometricparameters of the bearing are listed in Table 1.
Using these parameters, the characteristic frequenciescorresponding to different bearing faults were calculated us-
Table 1: Geometric parameters of the bearing.
Ball diameter d = 7.5 mm
Pitch diameter D = 39.45 mm
Contact angle α = 5 � 20◦
Number of rolling element (with two rows) n = 2× 13 = 26
0 10 20 30 40 50 60
Time (ms)
−20
−15
−10
−5
0
5
10
15
20
Am
plit
ude
Figure 6: Vibration signals of the ball bearing.
ing the following equations [14]:
f = n
2fr
(1− d
Dcosα
)for outer race failure,
f = n
2fr
(1 +
d
Dcosα
)for inner race failure,
f = D
dfr
[1−
(d
D
)2
cos2 α
]for rolling element failure,
(11)
where f stands for the characteristic frequency correspond-ing to different kinds of faults, fr =25 Hz the relative rotatingfrequency between the inner and the outer races, n the num-ber of rollers (balls), α the contact angle between the raceand the roller, d the roller diameter, andD the pitch diameterof the bearing. The theoretical characteristic failure frequen-cies calculated by (11) are between 263–267 Hz for outer racefailure, 383–387 Hz for inner race failure, and are 127 Hz fora ball flaw fault, respectively. The vibration signal collectedfrom the ball bearing is shown in Figure 6.
Let β = 5.8, ξ = 6, and by performing the complex Mor-let wavelet transform in the frequency region from 500 to6500 Hz, the results analyzed are shown in Figure 7.
It can be clearly seen from Figure 7 that some successiveimpulses are present in the whole frequency region. More-over, a time interval approximating to 8.6 milliseconds (cor-responding to a frequency value of 116 Hz) was found be-tween adjacent impulses, which was close to the characteris-tic frequency of 127 Hz for a ball flaw fault. It was suspectedtherefore that a ball fault had occurred in the bearing beinginspection.
In order to confirm this prediction, a physical inspection
Detecting Impulses in Mechanical Signals by Wavelets 1161
Ts = 8.6ms
Ts = 8.6ms
Ts = 8.6ms
Ts = 8.6ms
Ts = 8.6ms
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.11000
2000
3000
4000
5000
6000
Freq
uen
cy(H
z)
0 10 20 30 40 50 60
Time (ms)
Figure 7: Analyzed result of the signal shown in Figure 5.
Damage location
Figure 8: The damaged ball found in the inspected bearing.
of the bearing was undertaken and a damaged ball was found.It is as shown in Figure 8.
6. CONCLUSIONS
The above theoretical analysis and discussions lead to follow-ing conclusions.
(1) After processing the data using the proposed wavelet-based envelope analysis method, the impulses buried in thenoisy signals have been identified for further analysis. In theanalyzed results, the impulsive features become more explicitand much easier to identify, thus effectively facilitating thediagnosis of machine faults.
(2) With the aid of the adjustable decay parameter, theproposed soft-threshold function is more adaptive for featureextraction and can lead to a smoother result. It overcomes therigid performance of available hard/soft-threshold criteria.
(3) The advanced impulse detection technique devel-oped based on the proposed wavelet-based envelope analy-sis method and the new adaptive soft-threshold function iseffective at extracting the impulsive features from those me-chanical signals with low SNR.
ACKNOWLEDGMENTS
The work described in this paper was supported by the Na-tional Natural Science Fund of China (Ref. No. 50205021)and the Shaanxi Provincial Natural Science Fund (Ref. No.2002E226). The authors would like to express their specialappreciation to reviewers and Dr. M. D. Seymour. Their kindsuggestions significantly improved the quality of the paper.
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[12] P. Tse, G. Xu, L. S. Qu, and S. Kumara, “An effectiveand portable electronic stethoscope for fault diagnosis byanalysing machine running sound directly,” InternationalJournal of Acoustics and Vibration, vol. 6, no. 1, pp. 23–31,2001.
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1162 EURASIP Journal on Applied Signal Processing
W.-X. Yang got his Ph.D. degree from Xi’anJiaotong University in 1999. Now, he is anAssociate Professor working in Northwest-ern Polytechnical University, Xi’an, China.He majors in signal processing, nondestruc-tive detection, machine condition monitor-ing, fault diagnosis, and other related re-searches.
X.-M. Ren got his Ph.D. degree from North-western Polytechnical University in 1999.Now he is the Head of the Institute of Vibra-tion Engineering in this university. He ma-jors in vibration analysis, dynamics, signalprocessing, and automatic control.
